Enlargements of Quantales
-
Urmas Luhaäär
Abstract
In this paper, we study the enlargements of quantales. We prove three main results. First, if Q is a factorizable quantale, then any matrix quantale over Q is an enlargement of Q; second, any unital Rees matrix quantale over a quantale Q with an identity is an enlargement of Q; third, two quantales are Morita equivalent if and only if they have a joint enlargement. To prove these theorems, we use quantale matrices and modules and Morita contexts of quantales. Our main theorems and their proofs are parallel to those known for idempotent rings.
REFERENCES
[1] Abramsky, S.—Vickers, S.: Quantales, observational logic and process semantics, Math. Struct. Comp. Sci. 3 (1993), 161–227.10.1017/S0960129500000189Suche in Google Scholar
[2] Ánh, P. N.—Márki, L.: Rees matrix rings, J. Algebra 81 (1983), 340–369.10.1016/0021-8693(83)90193-XSuche in Google Scholar
[3] Borceux, F.—Vitale, E. M.: A Morita theorem in topology, Rend. Circ. Mat. Palermo (2) Suppl. 29 (1992), 353–362.Suche in Google Scholar
[4] Laan, V.—Väljako, K.: Enlargements of rings, Comm. Algebra 49 (2021), 1764–1772.10.1080/00927872.2020.1851702Suche in Google Scholar
[5] Lawson, M. V.: Enlargements of regular semigroups, Proc. Edinburgh Math. Soc. 39 (1996), 425–460.10.1017/S001309150002321XSuche in Google Scholar
[6] Lawson, M. V.: Morita equivalence of semigroups with local units, J. Pure Appl. Algebra 215 (2011), 455–470.10.1016/j.jpaa.2010.04.030Suche in Google Scholar
[7] Lawson, M. V.—Márki, L.: Enlargements and coverings by Rees matrix semigroups, Monatsh. Math. 129 (2000), 191–195.10.1007/s006050050070Suche in Google Scholar
[8] Márki, L.—Steinfeld, O.: A Rees matrix construction without regularity. Contributions to General Algebra 6, Hölder-Pichler-Tempsky, Vienna, 1988, pp. 197–202.Suche in Google Scholar
[9] Paseka, J.: Morita equivalence in the context of Hilbert modules. Proceedings of the Ninth Prague Topological Symposium, 2001, 223–251, Topol. Atlas, North Bay, ON, 2002.Suche in Google Scholar
[10] Paseka, J.: Characterization of Morita equivalence pairs of quantales, Internat. J. Theoret. Phys. 44 (2005), 875–883.10.1007/s10773-005-7065-8Suche in Google Scholar
[11] Paseka, J.: Morita equivalence for m-regular quantales. Contributions to General Algebra 16, Heyn, Klagenfurt, 2005, pp. 155–171.Suche in Google Scholar
[12] Petrich, M.: Rings and Semigroups, Springer-Verlag, 1974.10.1007/BFb0061470Suche in Google Scholar
[13] Petrich, M.: Some constructions related to Rees matrix rings, Acta Math. Univ. Comenian. (N.S.) 71 (2002), 69–91.Suche in Google Scholar
[14] Russo, C.: Quantale Modules, with Applications to Logic and Image Processing, PhD thesis, 2007; https://arxiv.org/pdf/0909.4493.pdf.Suche in Google Scholar
[15] Quantaloid, Nlab; https://ncatlab.org/nlab/show/quantaloid.Suche in Google Scholar
[16] Quantale, Encyclopedia of Mathematics; http://encyclopediaofmath.org/index.php?title=Quantale&oldid=51710.Suche in Google Scholar
[17] Rosenthal, K. I.: Quantales and Their Applications, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, 1990.Suche in Google Scholar
[18] Talwar, S.: Morita equivalence for semigroups, J. Austral. Math. Soc. (Series A) 59 (1995), 81–111.10.1017/S1446788700038489Suche in Google Scholar
[19] Talwar, S.: Strong Morita equivalence and a generalisation of the Rees theorem, J. Algebra 181 (1996), 371–394.10.1006/jabr.1996.0125Suche in Google Scholar
© 2023 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Remembering Professor Štefan Znám, 9.2.1936–17.7.1993
- Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
- Enlargements of Quantales
- Multiplicative Dependent Pairs in the Sequence of Padovan Numbers
- Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions
- On the Rational Parametric Solution of Diagonal Quartic Varieties
- Theory of Certain Non-Univalent Analytic Functions
- Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)
- A Conjecture on H3(1) for Certain Starlike Functions
- Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces
- Complete Monotonicity and Inequalities Involving the k-Gamma and k-Polygamma Functions
- Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms
- On a System of Difference Equations Defined by the Product of Separable Homogeneous Functions
- On the Existence of Bi-Lipschitz Equivalent Metrics in Semimetric Spaces
- The Lehmann Type II Teissier Distribution
- Asymptotic Predictive Inference of Negative Lower Tail Index Distributions
- On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
- The Rational Zero-Divisor Cup-Length of Oriented Partial Flag Manifolds
Artikel in diesem Heft
- Remembering Professor Štefan Znám, 9.2.1936–17.7.1993
- Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
- Enlargements of Quantales
- Multiplicative Dependent Pairs in the Sequence of Padovan Numbers
- Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions
- On the Rational Parametric Solution of Diagonal Quartic Varieties
- Theory of Certain Non-Univalent Analytic Functions
- Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)
- A Conjecture on H3(1) for Certain Starlike Functions
- Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces
- Complete Monotonicity and Inequalities Involving the k-Gamma and k-Polygamma Functions
- Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms
- On a System of Difference Equations Defined by the Product of Separable Homogeneous Functions
- On the Existence of Bi-Lipschitz Equivalent Metrics in Semimetric Spaces
- The Lehmann Type II Teissier Distribution
- Asymptotic Predictive Inference of Negative Lower Tail Index Distributions
- On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
- The Rational Zero-Divisor Cup-Length of Oriented Partial Flag Manifolds
